99-101] Superposition of Effects 91 



EI, E^ ... give rise to potentials F/, F 2 ', ..., then charges E 1 + E 1 ', E 2 + # 2 ', ... 

 will give rise to potentials Fi+ F/, F 2 + F 2 ', .... 



In words: if we superpose two systems of charges, the potentials produced 

 can be obtained by adding together the potentials corresponding to the two 

 component systems. 



Clearly the proposition can be extended so as to apply to the superposi- 

 tion of any number of systems. 



We can obviously deduce the following : 



// charges E lt E 2 , ... give rise to potentials F 1? F 2 , ..., then charges 



^ KE Zy ... give rise to potentials KV l} KV^ .... 



101. Suppose now that we have n conductors fixed in position and 

 uncharged. Let us refer to these conductors as conductor (1), conductor (2), 

 etc. Suppose that the result of placing unit charge on conductor (1) and 

 leaving the others uncharged is to produce potentials 



on the n conductors respectively, then the result of placing E 1 on (1) and 

 leaving the others uncharged is to produce potentials 



PuE,, pizE lt ...p ln Ei. 



Similarly, if placing unit charge on (2) and leaving the others uncharged 

 gives potentials 



>2i> Pv,---Pm, 



then placing E 2 on (2) and leaving the others uncharged gives potentials 



p 21 E 2 , pw E 2 , . . . p m E 2 . 



In the same way we can calculate the result of placing E 3 on (3), E 4 on 

 (4), and so on. 



If we now superpose the solutions we have obtained, we find that the 

 effect of simultaneous charges E l) E 2 , ... E n is to give potentials V lt F 2 , ... F w , 

 where 



etc 



These equations give the potentials in terms of the charges. The coeffi- 

 cients p n , p 2l , ... do not depend on either the potentials or charges, being 

 purely geometrical quantities, which depend on the size, shape and position 

 of the different conductors. 



